YES

Problem:
 strict:
  g(c(x,s(y))) -> g(c(s(x),y))
  f(c(s(x),y)) -> f(c(x,s(y)))
  f(f(x)) -> f(d(f(x)))
  f(x) -> x
 weak:
  rand(x) -> x
  rand(x) -> rand(s(x))

Proof:
 Matrix Interpretation Processor: dim=2
  
  interpretation:
             [1 0]  
   [d](x0) = [0 0]x0,
   
                 [1 1]     [1 0]     [1]
   [c](x0, x1) = [2 2]x0 + [1 0]x1 + [2],
   
               
   [f](x0) = x0,
   
                [2 0]     [2]
   [rand](x0) = [0 1]x0 + [0],
   
               
   [s](x0) = x0,
   
             [2 0]     [1]
   [g](x0) = [1 1]x0 + [0]
  orientation:
                  [2 2]    [2 0]    [3]    [2 2]    [2 0]    [3]               
   g(c(x,s(y))) = [3 3]x + [2 0]y + [3] >= [3 3]x + [2 0]y + [3] = g(c(s(x),y))
   
                  [1 1]    [1 0]    [1]    [1 1]    [1 0]    [1]               
   f(c(s(x),y)) = [2 2]x + [1 0]y + [2] >= [2 2]x + [1 0]y + [2] = f(c(x,s(y)))
   
                  [1 0]              
   f(f(x)) = x >= [0 0]x = f(d(f(x)))
   
                    
   f(x) = x >= x = x
   
             [2 0]    [2]         
   rand(x) = [0 1]x + [0] >= x = x
   
             [2 0]    [2]    [2 0]    [2]             
   rand(x) = [0 1]x + [0] >= [0 1]x + [0] = rand(s(x))
  problem:
   strict:
    g(c(x,s(y))) -> g(c(s(x),y))
    f(c(s(x),y)) -> f(c(x,s(y)))
    f(f(x)) -> f(d(f(x)))
    f(x) -> x
   weak:
    rand(x) -> rand(s(x))
  Bounds Processor:
   bound: 1
   enrichment: match-rt
   automaton:
    final states: {7}
    transitions:
     g0(7) -> 7*
     s1(14) -> 14*
     s1(7) -> 14*
     d1(7) -> 15*
     s0(7) -> 7*
     c0(7,7) -> 7*
     c1(14,14) -> 45*
     c1(14,7) -> 45*
     c1(7,14) -> 15*
     g1(45) -> 7*
     f0(7) -> 7*
     rand0(7) -> 7*
     d0(7) -> 7*
     rand1(14) -> 7*
     f1(7) -> 7*
     f1(15) -> 7*
     15 -> 7*
   problem:
    strict:
     g(c(x,s(y))) -> g(c(s(x),y))
     f(f(x)) -> f(d(f(x)))
     f(x) -> x
    weak:
     rand(x) -> rand(s(x))
   Bounds Processor:
    bound: 1
    enrichment: match-rt
    automaton:
     final states: {7}
     transitions:
      g0(7) -> 7*
      s1(10) -> 10*
      s1(45) -> 46*
      s1(7) -> 10*
      d1(25) -> 26*
      d1(38) -> 39*
      s0(7) -> 7*
      c0(7,7) -> 7*
      c1(10,7) -> 11*
      g1(11) -> 7*
      f0(7) -> 7*
      rand0(7) -> 7*
      d0(7) -> 7*
      rand1(36) -> 37*
      f1(26) -> 27*
      f1(24) -> 25*
      27 -> 25,38,7
      24 -> 25*
      46 -> 36*
      7 -> 24*
      36 -> 45*
      26 -> 27*
      37 -> 7*
      39 -> 26*
      10 -> 36*
    problem:
     strict:
      f(f(x)) -> f(d(f(x)))
      f(x) -> x
     weak:
      rand(x) -> rand(s(x))
    Arctic Interpretation Processor:
     dimension: 1
     interpretation:
      [d](x0) = x0,
      
      [f](x0) = 1x0,
      
      [rand](x0) = 4x0,
      
      [s](x0) = x0
     orientation:
      f(f(x)) = 2x >= 2x = f(d(f(x)))
      
      f(x) = 1x >= x = x
      
      rand(x) = 4x >= 4x = rand(s(x))
     problem:
      strict:
       f(f(x)) -> f(d(f(x)))
      weak:
       rand(x) -> rand(s(x))
     Bounds Processor:
      bound: 2
      enrichment: match-rt
      automaton:
       final states: {5}
       transitions:
        s1(28) -> 29*
        d1(11) -> 12*
        d1(14) -> 15*
        s0(5) -> 5*
        f0(5) -> 5*
        rand2(32) -> 33*
        rand0(5) -> 5*
        d0(5) -> 5*
        f2(22) -> 23*
        f2(20) -> 21*
        rand1(29) -> 30*
        d2(21) -> 22*
        s2(31) -> 32*
        s2(38) -> 39*
        f1(12) -> 13*
        f1(10) -> 11*
        30 -> 5*
        12 -> 20*
        13 -> 11,14,5
        5 -> 28,10
        39 -> 32*
        15 -> 12*
        29 -> 31*
        32 -> 38*
        23 -> 11*
        33 -> 30*
      problem:
       strict:
        
       weak:
        rand(x) -> rand(s(x))
      Qed